“The pigsty that is Washington, D.C., gets mud on a lot of people and the question is what are you going to do moving forward.”

Rep. Jason Chaffetz, R-Utah

WASHINGTON — President Barack Obama summoned congressional leaders to the White House on Wednesday as a partial government shutdown entered a second day with little sign of a breakthrough to get hundreds of thousands of people back to work. Some on Capitol Hill ominously suggested the impasse might last for weeks, but a few Republicans seemed ready to blink.

House Speaker John Boehner's office said the Ohio Republican would attend the White House meeting Wednesday afternoon, casting it as a sign the president is ready to start negotiating on GOP demands to extract changes to the new health care law in exchange for funding the government.

"We're pleased the president finally recognizes that his refusal to negotiate is indefensible,"" Boehner spokesman Brendan Buck said. "It's unclear why we'd be having this meeting if it's not meant to be a start to serious talks between the two parties."

Senate Majority Leader Harry Reid, D-Nev., Senate Minority Mitch McConnell, R-Ky., and House Minority Leader Nancy Pelosi, D-Calif., were also to attend the White House meeting. An Obama adviser said Obama would urge House Republicans to pass a spending bill free of other demands.

Some Republican lawmakers appear ready to take that step. Republican Rep. Peter King of New York accused tea party-backed lawmakers of trying to "hijack the party" and said he senses that a growing number of rank-and-file House Republicans — perhaps as many as a hundred — are tired of the shutdown that began Tuesday morning and will be meeting to look for a way out.

But GOP leaders and tea party-backed members seemed determined to press on. The House GOP leadership announced plans to continue trying to open more popular parts of the government. They planned to pass five bills to open national parks, processing of veterans' claims, the Washington, D.C., government, medical research, and to pay members of the National Guard.

The White House immediately promised a veto, saying opening the government on a piecemeal basis is unacceptable.

"Instead of opening up a few government functions, the House of Representatives should re-open all of the government," the White House said in an official policy statement.

The move presented Democrats with politically challenging votes but they rejected the idea, saying it was unfair to pick winners and losers as federal employees worked without a guarantee of getting paid and the effects of the partial shutdown rippled through the country and the economy.

Funding for much of the U.S. government was halted after Republicans hitched a routine spending bill to their effort to kill or delay the health care law they call "Obamacare." The president accuses them of holding the government hostage.

Rep. Jason Chaffetz, a tea party favorite, said there would be no solution until President Barack Obama and Democrats who control the Senate agree to discuss problems with the nation's unfolding health care overhaul.

"The pigsty that is Washington, D.C., gets mud on a lot of people and the question is what are you going to do moving forward," Chaffetz, R-Utah, said on CBS' "This Morning."

Meanwhile, another financial showdown even more critical to the economy was looming. Treasury Secretary Jacob Lew told Congress that unless lawmakers act in time, he will run out of money to pay the nation's bills by Oct. 17. Congress must periodically raise the limit on government borrowing to keep U.S. funds flowing, a once-routine matter that has become locked in battles over the federal budget deficit.

Rep. Steny Hoyer of Maryland, the second-ranking House Democrat, said Democrats would overwhelmingly accept a short-term spending measure to reopen the government and increase the nation's debt limit while other political differences are worked out. "That would be a responsible way to go," Hoyer told CNN.